## Displaying similar documents to “Regularity of Gaussian white noise on the d-dimensional torus”

### Fermat test with Gaussian base and Gaussian pseudoprimes

Czechoslovak Mathematical Journal

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The structure of the group ${\left(ℤ/nℤ\right)}^{☆}$ and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group ${𝒢}_{n}:=\left\{a+b\mathrm{i}\in ℤ\left[\mathrm{i}\right]/nℤ\left[\mathrm{i}\right]:{a}^{2}+{b}^{2}\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}n\right)\right\}$. In particular, we characterize Gaussian Carmichael...

### Note on the variance of the sum of Gaussian functionals

Applicationes Mathematicae

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Let $\left({X}_{i},i=1,2,...\right)$ be a Gaussian sequence with ${X}_{i}\in N\left(0,1\right)$ for each i and suppose its correlation matrix $R={\left({\rho }_{ij}\right)}_{i,j\ge 1}$ is the matrix of some linear operator R:l₂→ l₂. Then for ${f}_{i}\in L²\left(\mu \right)$, i=1,2,..., where μ is the standard normal distribution, we estimate the variation of the sum of the Gaussian functionals ${f}_{i}\left({X}_{i}\right)$, i=1,2,... .

### Note on the multidimensional Gebelein inequality

Applicationes Mathematicae

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We generalize the Gebelein inequality for Gaussian random vectors in ${ℝ}^{d}$.

### Gebelein's inequality and its consequences

Banach Center Publications

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Let $\left({X}_{i},i=1,2,...\right)$ be the normalized gaussian system such that ${X}_{i}\in N\left(0,1\right)$, i = 1,2,... and let the correlation matrix ${\rho }_{ij}=E\left({X}_{i}{X}_{j}\right)$ satisfy the following hypothesis: $C=su{p}_{i\ge 1}{\sum }_{j=1}^{\infty }|{\rho }_{i,j}|<\infty$. We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

### Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $Hgt;1/3$. We show that under some geometric conditions, in the regular case $Hgt;1/2$, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $Hgt;1/3$ and under the same geometric conditions, we show that the density of the solution is smooth and...

### Convergence rates for the full gaussian rough paths

Annales de l'I.H.P. Probabilités et statistiques

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Under the key assumption of finite $\rho$-variation, $\rho \in \left[1,2\right)$, of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), $\rho =1$ resp. $\rho =1/\left(2H\right)$, we recover and extend the respective results of ( (2009) 2689–2718) and ( (2012) 518–550). In particular, we establish an a.s. rate ${k}^{-\left(1/\rho -1/2-\epsilon \right)}$, any $\epsilon gt;0$, for Wong–Zakai and Milstein-type approximations...

### Sub-Laplacian with drift in nilpotent Lie groups

Colloquium Mathematicae

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We consider the heat kernel ${\varphi }_{t}$ corresponding to the left invariant sub-Laplacian with drift term in the first commutator of the Lie algebra, on a nilpotent Lie group. We improve the results obtained by G. Alexopoulos in [1], [2] proving the “exact Gaussian factor” exp(-|g|²/4(1+ε)t) in the large time upper Gaussian estimate for ${\varphi }_{t}$. We also obtain a large time lower Gaussian estimate for ${\varphi }_{t}$.

### Optimal Constants in Khintchine Type Inequalities for Fermions, Rademachers and q-Gaussian Operators

Bulletin of the Polish Academy of Sciences. Mathematics

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For $\left({P}_{k}\right)$ being Rademacher, Fermion or q-Gaussian (-1 ≤ q ≤ 0) operators, we find the optimal constants ${C}_{2n}$, n∈ ℕ, in the inequality $\parallel {\sum }_{k=1}^{N}{A}_{k}\otimes {P}_{k}{\parallel }_{2n}\le {\left[{C}_{2n}\right]}^{1/2n}max{\parallel \left({\sum }_{k=1}^{N}A{*}_{k}{A}_{k}}^{1/2}{\parallel }_{{L}_{2n}},\parallel \left({\sum }_{k=1}^{N}{A}_{k}A{*}_{k}$1/2∥L2n$,$valid for all finite sequences of operators $\left({A}_{k}\right)$ in the non-commutative ${L}_{2n}$ space related to a semifinite von Neumann algebra with trace. In particular, ${C}_{2n}=\left(2nr-1\right)!!$ for the Rademacher and Fermion sequences.

### Stationary distributions for jump processes with memory

Annales de l'I.H.P. Probabilités et statistiques

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We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$. The state space of $\left(Z,S\right)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $\left(Z,S\right)$ is the product of the uniform probability measure and a Gaussian distribution.

### Stein’s method in high dimensions with applications

Annales de l'I.H.P. Probabilités et statistiques

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Let $h$ be a three times partially differentiable function on ${ℝ}^{n}$, let $X=\left({X}_{1},...,{X}_{n}\right)$ be a collection of real-valued random variables and let $Z=\left({Z}_{1},...,{Z}_{n}\right)$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $𝔼h\left(X\right)-𝔼h\left(Z\right)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\to \infty$. In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic...

### Iterated quasi-arithmetic mean-type mappings

Colloquium Mathematicae

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We work with a fixed N-tuple of quasi-arithmetic means $M₁,...,{M}_{N}$ generated by an N-tuple of continuous monotone functions $f₁,...,{f}_{N}:I\to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping ${I}^{N}\ni b↦\left(M₁\left(b\right),...,{M}_{N}\left(b\right)\right)$ tend pointwise to a mapping having values on the diagonal of ${I}^{N}$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M₁,...,{M}_{N}$ taken...

### Infinite dimensional Gegenbauer functionals

Banach Center Publications

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he paper is devoted to investigation of Gegenbauer white noise functionals. A particular attention is paid to the construction of the infinite dimensional Gegenbauer white noise measure ${}_{\beta }$, via the Bochner-Minlos theorem, on a suitable nuclear triple. Then we give the chaos decomposition of the L²-space with respect to the measure ${}_{\beta }$ by using the so-called β-type Wick product.

### An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes

Annales de la faculté des sciences de Toulouse Mathématiques

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We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval $\left[0,T\right]$ representing a chromosome. The originality is in the fact that some genotypes are missing. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on $\left[0,T\right]$ and under local alternatives with a QTL at ${t}^{☆}$ on $\left[0,T\right]$. We show that the LRT process is asymptotically...

### Gap universality of generalized Wigner and $\beta$-ensembles

Journal of the European Mathematical Society

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We consider generalized Wigner ensembles and general $\beta$-ensembles with analytic potentials for any $\beta \ge 1$. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian $\beta$-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal....

### On the strong Brillinger-mixing property of $\alpha$-determinantal point processes and some applications

Applications of Mathematics

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First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C\left(x,y\right)$ defining an $\alpha$-determinantal point process (DPP). Assuming absolute integrability of the function ${C}_{0}\left(x\right)=C\left(o,x\right)$, we show that a stationary $\alpha$-DPP with kernel function ${C}_{0}\left(x\right)$ is “strongly” Brillinger-mixing, implying, among others, that its tail-$\sigma$-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...

### The majorizing measure approach to sample boundedness

Colloquium Mathematicae

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We describe an alternative approach to sample boundedness and continuity of stochastic processes. We show that the regularity of paths can be understood in terms of the distribution of the argument maximum. For a centered Gaussian process X(t), t ∈ T, we obtain a short proof of the exact lower bound on $su{p}_{t\in T}X\left(t\right)$. Finally we prove the equivalence of the usual majorizing measure functional to its conjugate version.

### Stable random fields and geometry

Banach Center Publications

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Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for $M={S}^{m}$, the m-dimensional sphere. Let $Y\left(B\right);B\in ℬ\left({S}^{m}\right)$ be the Gaussian random measure on ${S}^{m}$, that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on ${S}^{m}$, 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for ${B}_{i}$, i = 1,2,..., ${B}_{i}\cap {B}_{j}=\varnothing$,...

### Multilinear Fourier multipliers with minimal Sobolev regularity, I

Colloquium Mathematicae

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We find optimal conditions on m-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces ${H}^{{p}_{k}}$, $0<{p}_{k}\le 1$, to Lebesgue spaces ${L}^{p}$. These conditions are expressed in terms of L²-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky (1977) and in the bilinear case (m = 2) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral...

### Pointwise regularity associated with function spaces and multifractal analysis

Banach Center Publications

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The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces ${C}_{E}^{\alpha }\left(x₀\right)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E={L}^{\infty }$ corresponds to the usual Hölder regularity,...

### Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

Studia Mathematica

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Let ε > 0 and 1 ≤ k ≤ n and let ${{W}_{l}}_{l=1}^{p}$ be affine subspaces of ℝⁿ, each of dimension at most k. Let $m=O\left({\epsilon }^{-2}\left(k+logp\right)\right)$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H:ℝⁿ\to {ℝ}^{m}$ such that for all 1 ≤ l ≤ p and $x,y\in {W}_{l}$ we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces...

### Estimating composite functions by model selection

Annales de l'I.H.P. Probabilités et statistiques

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We consider the problem of estimating a function $s$ on ${\left[-1,1\right]}^{k}$ for large values of $k$ by looking for some best approximation of $s$ by composite functions of the form $g\circ u$. Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions $g,u$ and statistical frameworks. In particular, we handle the problems of approximating $s$ by additive functions, single and multiple index models, artificial neural networks, mixtures...

### Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces

Studia Mathematica

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We collect and extend results on the limit of ${\sigma }^{1-k}{\left(1-\sigma \right)}^{k}{|v|}_{l+\sigma ,p,\Omega }^{p}$ as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and ${|·|}_{l+\sigma ,p,\Omega }$ is the intrinsic seminorm of order l+σ in the Sobolev space ${W}^{l+\sigma ,p}\left(\Omega \right)$. In general, the above limit is equal to $c{\left[v\right]}^{p}$, where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.